Finite Group Databasebeta

← #307989

Subgroups — #307989

Order 512.

623 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26512C211yesmaximal, characteristic, complemented
2#26045C211yesmaximal, characteristic
3#22963C211yesmaximal, characteristic
4#22705C211yesmaximal, characteristic
5#10684C211yesmaximal, characteristic, complemented
6#10656C211yesmaximal, characteristic, complemented
7#10589C211yesmaximal, characteristic, complemented
8#10579C211yesmaximal, characteristic, complemented
9#9611C211yesmaximal, characteristic, complemented
10#9536C211yesmaximal, characteristic, complemented
11#9215C211yesmaximal, characteristic
12#8973C211yesmaximal, characteristic
13#8929C211yesmaximal, characteristic, complemented
14#8882C211yesmaximal, characteristic
15#8845C211yesmaximal, characteristic
16C2×#478—21no—
17C2×#476C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19#2552C2211yescharacteristic
20#2369C2211yescharacteristic
21D4×C4⋊C4C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23#2100—21no—
24#2088—21no—
25#2087—21no—
26#2059—21no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1987C2211yescharacteristic
30#1956C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947—21no—
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1941—21no—
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1921—21no—
41#1919—21no—
42#1864—21no—
43#1864—21no—
44#1852—21no—
45#1852—21no—
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1834C2211yescharacteristic
49#1808—21no—
50#1802C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×#125—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75#478—44no—
76#476—22no—
77#476Q814yes—
78C8×D4—22no—
79C8×D4—22no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—41no—
85C2×#107D411yescharacteristic
86C2×#107D411yescharacteristic
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—41no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102D411yescharacteristic
101C2×#102D411yescharacteristic
102C2×C22⋊C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#398—41no—
108#391—41no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#385—41no—
116#385—41no—
117#384—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#340D411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#334—22no—
150#333—22no—
151D4⋊C8—22no—
152C22×Q8—21no—
153C22×D4C2×D411yescharacteristic
154C23×C4—21no—
155C23×C4—41no—
156C23×C4—41no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—24no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—81no—
164C22×C8—81no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8C2×Q812yes—
173#125—42no—
174#125—81no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—81no—
180C4×D4—81no—
181C4×D4C2×Q812yes—
182C4×D4#4112yes—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—81no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—41no—
218C2×C42C2411yescharacteristic
219#107—22no—
220#107—41no—
221#107—42no—
222#107—42no—
223#107—81no—
224#107—81no—
225#107—82no—
226#106—24no—
227#106—42no—
228#106—81no—
229#106—81no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8C2×Q812yes—
239C4⋊C8#4112yes—
240#103—82no—
241#102—24no—
242#102—22no—
243#102—42no—
244#102Q1614yes—
245C22⋊C8—24no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—81no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—81no—
265C2×Q8—81no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4C22⋊Q812yes—
270C2×D4C2×Q1612yes—
271C2×D4AGL(1,ℤ/8)12yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—161no—
294C22×C4—161no—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4H(5,𝔽2)11yescharacteristic
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—84no—
325C2×C8C2×Q1612yes—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—82no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—161no—
361C4⋊C4—161no—
362C4⋊C4C22⋊Q812yes—
363C4⋊C4C2×Q1612yes—
364C4⋊C4#13512yes—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—161no—
376C42C22×Q811yescharacteristic
377C42C2×#4111yescharacteristic
378C42C2×#4111yescharacteristic
379C422−1+411yescharacteristic
380C23—22no—
381C23—22no—
382C23—22no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23D4211yescharacteristic
388Q8—82no—
389Q8—161no—
390D4—22no—
391D4—22no—
392D4—22no—
393D4#47714yes—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—164no—
434C2×C4—164no—
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4#52511yescharacteristic
437C2×C4#53911yescharacteristic
438C2×C4#54711yescharacteristic
439C2×C4D4×Q811yescharacteristic
440C2×C4#55411yescharacteristic
441C2×C4C22×Q1611yescharacteristic
442C2×C4C2×#13511yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
444C2×C4#57511yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57811yescharacteristic
449C8—22no—
450C8—42no—
451C8—42no—
452C8—41no—
453C8—41no—
454C8—41no—
455C8—42no—
456C8—81no—
457C8—81no—
458C22—24no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22#244911yescharacteristic
466C22#326411yescharacteristic
467C22#326711yescharacteristic
468C22#328211yescharacteristic
469C22D4×Q1611yescharacteristic
470C22#328811yescharacteristic
471C22#329111yescharacteristic
472C4—21no—
473C4—21no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—82no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4C2×#47711yescharacteristic
486C4#308311yescharacteristic
487C4#310211yescharacteristic
488C4#312311yescharacteristic
489C4#333811yescharacteristic
490C4#334311yescharacteristic
491C4#338111yescharacteristic
492C4#338811yescharacteristic
493C2—44no—
494C2#1978811yescharacteristic
495C2#1981411yescharacteristic
496C2#2000111yescharacteristic
497C2#2002611yescharacteristic
498C2#2634211yescharacteristic
499C2#2637811yescharacteristic
500C2#2640711yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26512C2semidirect
5. #10684C2semidirect
6. #10656C2semidirect
7. #10589C2semidirect
8. #10579C2semidirect
9. #9611C2semidirect
10. #9536C2semidirect
13. #8929C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#265122
#260452
#229632
#227052
#106842
#106562
#105892
#105792
#96112
#95362
#92152
#89732
#89292
#88822
#88452